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Mathematics, Scienceand Technology

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Chapter

6

138

Science and technology have left their mark on our wayof life and our environment and are among the mostrevealing examples of human ingenuity. Scientific discov-eries and technological achievements affect majoraspects of our existence. Computers, for example, haverevolutionized the way we work and communicate, andeven the way we think, and have become the principaltool for acquiring knowledge in many fields.

Moreover, science and technology would not havereached their current level of development without thecontribution of mathematics. Although each subject areafollowed its own course of development, they havebecome more closely related as they have evolved. Moreoften than not, technical objects with any measure ofsophistication work by making use of components thatoperate according to the principles of mathematical logic.However, widespread use of mathematics has not beenlimited to the fields of science and technology. Countlesssituations require us to decode numerical information,estimate, calculate and measure, all of which are opera-tions that belong to the world of mathematics.

Mathematics, science and technology each developaccording to their own dynamic, but this development isalso a function of the external pressure exerted by a soci-ety seeking ways to meet some of its needs.Mathematical developments, scientific discoveries andtechnological achievements must be placed in their his-torical, social, economic and cultural context if we are tounderstand how these three areas have evolved.

For the most part, scientific and technological advancescontribute to our individual and collective well-being, butsome of these advances may also threaten the ecologicalbalance of our environment or introduce new elementswhose long-term environmental effects are difficult toforesee. Only by acquiring a broad general knowledge ofscience and technology will students be able to take acritical look at these changes and appreciate the ethicalissues they raise.

GENERAL OBJECTIVE IN MATHEMATICS, SCIENCE ANDTECHNOLOGY

To provide access to a specific set of knowledges relatedto the methods, conceptual fields and languages specificto each of the subjects in this subject area.

CORE LEARNINGS IN MATHEMATICS, SCIENCE ANDTECHNOLOGY

◗ Understands information and conveys it clearly usinglanguage appropriate to mathematics, science andtechnology: terminology, graphics, notation, symbolsand codes

◗ Uses inductive and deductive reasoning

◗ Establishes connections between the learningshe/she acquires in each subject in this particular sub-ject area and the learnings related to other subjects

◗ Views this knowledge as a tool that can be used ineveryday life

◗ Analyzes data resulting from observations or foundin a problem and uses appropriate strategies toachieve a result or to find a solution that can then beexplained, verified, interpreted and generalized

◗ Appreciates the importance of mathematics, scienceand technology in human history

◗ Exercises critical judgment in assessing the impact ofmathematics, science and technology on individuals,society and the environment

Chapter 6 Mathematics, Science and Technology

1391396.1 MathematicsMathematics, Science and Technology

Mathematics 140Mathematics, Science and Technology

Mathematics is a major source of intellectual develop-ment and a determining factor in students’ educationalsuccess. Its mastery is also a significant asset when itcomes to carving out a place for oneself in a societywhere the practical applications of mathematics are asnumerous as they are varied. High technology, engineer-ing and computer programming are among the manyfields requiring the use of mathematics, but it is also usedin manufacturing common everyday objects, in measur-ing time or in organizing space.

Mathematics involves abstraction. Although it is alwaysto the teacher’s advantage to refer to real-world objectsand situations, he/she must nevertheless set out to exam-ine, in the abstract, relationships between the objects orbetween the elements of a given situation. For example,a triangular object becomes a geometric figure, andtherefore a subject of interest to mathematicians, as soonas we begin to study the relationships between its sides,its vertices and its angles, for example.

The program is organized around three competencies: thefirst refers to the ability to solve situational problems; thesecond pertains to mathematical reasoning, whichimplies familiarity with concepts and processes specificto mathematics; and the third focuses on communicationusing mathematical language.

Mathematical activities always involve the examinationof situational problems. The process of solving situational

problems is a topic in and of itself, but problem solving isalso an instructional tool that can be used in the vastmajority of mathematical learning processes. It is of par-ticular importance because the cognitive activity associ-ated with mathematics involves logical reasoning appliedto situational problems.

Reasoning in mathematics consists in establishing rela-tionships, combining them and using them to perform avariety of operations in order to create new concepts andtake one’s mathematical thinking to a higher level. In ele-mentary school, students develop the ability to engage indeductive, inductive and creative mathematical reason-ing.They become familiar with deductive reasoning whenlearning how to draw a conclusion from the informationgiven in a situational problem. They become familiar withinductive reasoning when asked to derive rules or lawson the basis of their observations. They become familiarwith creative reasoning because they must devise combi-nations of operations in order to find different solutionsto a situational problem.

Communication using mathematical language servestwo purposes: to familiarize the students with mathe-matical terminology and to teach them about the justifi-cation process. In the first case, the students discover newwords and new meanings for known words. In the sec-ond case, they learn how to provide complete and preciseexplanations for a procedure or a line of reasoning.

Mastery of mathematics is a

significant asset when it comes

to carving out a place for one-

self in a society where its practi-

cal applications are as numer-

ous as they are varied.

Introduction


On a different level, incorporating a historical dimensioninto the mathematics curriculum is an excellent way ofenhancing its cultural component. This provides studentswith the opportunity to understand the evolution, mean-ing and usefulness of mathematics and to discover thatthe invention and development of certain instrumentssuch as the ruler, the abacus, the protractor and the cal-culator were directly or indirectly related to practicalneeds that emerged in different societies. An overview ofhistory can also illustrate the fact that mathematicalknowledge results from the extensive work of mathe-maticians with a passion for their subject.

Lastly, technology can prove to be a valuable tool thatwill help the students solve situational problems, under-stand concepts and processes and carry out assignedtasks more efficiently.

The development of the three competencies covered inthe program is closely connected with the acquisition ofknowledges related to arithmetic, geometry, measure-ment, statistics and probability. These branches of math-ematics include the mathematical concepts andprocesses studied in this program. In essence, the threecompetencies are distinguished by the emphasis placedon different facets of mathematical thinking, which are,in fact, all integrated. Such a distinction should make iteasier to understand this thinking and to structure thepedagogical process, but it in no way suggests that theseelements should be examined separately. Logicallyspeaking, we cannot reason using mathematical con-cepts and processes without communicating in mathe-matical language, and we usually engage in mathemati-cal reasoning when solving situational problems.

Figure 8Mathematics


MEANING OF THE COMPETENCY

The capacity to solve a situational problem is an intellec-tual process used in a wide variety of situations. On apractical level, it is spontaneously used to meet variouseveryday challenges. On a more abstract level, it canprove to be a powerful intellectual tool that develops rea-soning and creative intuition. It can be as useful to thosewho wish to understand or resolve theoretical and con-ceptual enigmas as it is to a statistician whose work hasimmediate practical consequences. Relatively speaking, itis also useful to a student who is asked to find a way ofdetermining the number of objects in a collection or cal-culating the area of a rectangle.

When solving situational problems in preschool and ele-mentary school, students are engaged in a process thatinvolves using different strategies related to comprehen-sion, organization, problem solving, validation and com-munication. These problems also provide an opportunityto employ mathematical reasoning and to communicateusing mathematical language.

CONNECTIONS TO CROSS-CURRICULAR COMPETENCIES

Because of its scope, the competency of solving a situa-tional problem makes it possible to develop all the cross-curricular competencies. In particular, it requires studentsto use creativity and encourages them to process infor-mation, find efficient ways of working (often in teams)

and develop appropriate ways of communicating. In allthese respects, there is considerable overlap between thiscompetency and the cross-curricular competency dealingwith the ability to solve problems.

CONTEXT FOR LEARNING

A situational problem is characterized by the need toattain a goal, carry out a task or find a solution. Thisobjective cannot be instantly achieved, since it is not anexercise involving applications. On the contrary, itsachievement requires reasoning, research and the use ofstrategies to mobilize learnings. When solving situationalproblems in mathematics, the students must also performa series of operations in order to decode, model, verify,explain and validate. This is a dynamic process thatinvolves anticipating results, redoing certain steps in theproblem-solving procedure and exercising critical judg-ment.

A situational problem relates to a specific context andprovides a challenge geared to the students’ capabilities.It must arouse and engage their interest and encouragethem to take action in order to work out a solution.Lastly, it must involve some concern for metacognitivereflection.

Situational problems can require the use of arithmetic,geometry, measurement, statistics and probability. Theymay deal with purely mathematical questions or practical

questions that are in some way familiar to the studentsand that are related to actual or realistic situations.Depending on the purpose of these problems, they mayinvolve dealing with complete, superfluous, implicit ormissing information.

DEVELOPMENTAL PROFILE

During Cycle One, the students learn how to identify therelevant information in a situational problem. They seehow the information given in the situational problemrelates to the assigned task.They also learn how to modela situational problem, apply different strategies and rec-tify their solution in light of their results and discussionswith their classmates.

During Cycle Two, the students succeed in identifying theimplicit information in situational problems and increasetheir ability to develop models and apply a variety ofstrategies. They can describe the procedure they haveused and explain how they went about their work, andthey may be interested in approaches that differ fromtheir own.

During Cycle Three, the students are able to decode situ-ational problems that involve recognizing situations inwhich information is missing. They work more indepen-dently in developing models and find it easier to devisestrategies. They are better at validating their solution andcommenting on their classmates’ solutions.

COMPETENCY 1 • TO SOLVE A SITUATIONAL PROBLEM RELATED TO MATHEMATICS

Focus of the Competency

CYCLE ONE

By the end of this cycle, the students solve a situationalproblem based on complete information. They determinethe task to be performed and find the relevant informa-tion by using different types of representations such asobjects, drawings, tables, graphs, symbols or words. Theywork out a solution involving one or two steps and occa-sionally check the result. Using basic mathematical lan-guage, they explain their solution (procedure and finalanswer) orally or in writing.

CYCLE TWO

By the end of this cycle, the students solve a situationalproblem that may involve more than one type of infor-mation. They are more careful in choosing the types ofrepresentations they will use to highlight the relevantinformation in the situational problem, and they may alsouse diagrams. They anticipate the result and work out asolution involving a few steps. They validate the solution(procedure and final answer) and explain it orally or inwriting using elaborate mathematical language.

CYCLE THREE

By the end of this cycle, the students solve a situationalproblem involving different types of information. Theymake more appropriate use of the various types of repre-sentations that allow them to organize this information.They anticipate the result, work out a solution that mayinvolve several steps, and associate the presentation ofthe problem with that of similar problems. They validatethe solution (procedure and final answer) and explain itorally or in writing using exact mathematical language.


End-of-Cycle Outcomes

Evaluation Criteria– Production of a correct solution

(procedure and final answer) ➊ ➋ ➌

– Oral or written explanation of themain aspects of the solution ➊ ➋ ➌

– Appropriate oral or written explanationof how the solution was validated ➋ ➌

Legend:* ➊ Cycle One ➋ Cycle Two ➌ Cycle Three

* This legend also applies to the Evaluation Criteria for the othercompetencies and to the sections entitled Essential Knowledgesand Suggestions for Using Information and CommunicationsTechnologies.

Key Features of the Competency

To model the situational problem

To apply differentstrategies to work outa solution

To validate the solution

To share informationrelated to the solution

TO SOLVE A

SITUATIONAL PROBLEM

RELATED TO

MATHEMATICS

To decode the elements ofthe situational problem



To reason is to logically organize a series of facts, ideas orconcepts in order to arrive at a conclusion that should bemore reliable than one resulting from an impression orintuition. This does not mean that there is no place forintuition and creativity, but these faculties must be chan-nelled into efforts that ultimately lead to a clearly statedconclusion based on a certain line of reasoning.

In mathematics, organizing means engaging in mentalactivities such as abstracting, coordinating, differentiat-ing, integrating, constructing and structuring.These activ-ities, which deal with the relationships between objectsor between their components, should, for example, helpthe students understand the additive and multiplicativeproperties of a number or its ordinal and cardinal dimen-sions. These activities can also help them discover themeaning of iteration as it pertains to measurements, ofequality or inequality in an equation and of direct orinverse proportionality.

Mathematical reasoning involves apprehending the situ-ation, mobilizing relevant concepts and processes andmaking connections. In so doing, the students becomefamiliar with mathematical language, construct themeaning of mathematical concepts and processes andestablish links between them. This approach also encour-ages the students to use mathematical instruments.

Different examples illustrate how this competency isused. In arithmetic, the students construct the meaningof numbers, number systems and operations. In geome-try, they discover the characteristics of plane figures andsolids and establish spatial relationships. With regard tomeasurement, they study what measurements, units ofmeasure and their interrelationships mean. With regardto probability, they examine random events by, for exam-ple, formulating their conclusions in terms of whetherthese events are certainties, possibilities or impossibili-ties. In statistics, they interpret and draw graphs repre-senting various aspects of everyday life.

With respect to processes, the students spontaneouslydevise their own ways of doing things by using instru-ments or technology, and explore these methods in orderto understand how they work. For example, instead ofusing recognized algorithms, the students can begin bycarrying out arithmetic operations based on relativelyunstructured intuition. Measurements can be taken usingany object as a unit of measurement. However, mathe-matics has its own processes and instruments that havebecome well-established conventions over time. In addi-tion, when it comes to learning about instruments, theinstructional goal should be to ensure that the studentsare able to use these conventional tools intelligently andto understand what they are doing, while developingtheir measurement sense.


When the students use mathematical concepts andprocesses, they also develop cross-curricular competen-cies, especially intellectual competencies relating to theexercise of critical judgment and the use of creativity.They also use the methodological competency relating tothe development of effective work methods, as well asthe communication-related competency.


Fostering the development of this competency involvesusing situational problems that will force students to askthemselves questions, to make connections between theelements they are examining and to look for answers totheir questions. These situations deal with arithmetic,geometry, measurement, statistics and probability andoccasionally relate to the history of mathematics.

The students primarily use manipulative materials, makeuse of technology and consult a resource person, if nec-essary. They use tools ranging from an ordinary piece ofgraph paper to a computer. When they use processesthat call for specific instruments (e.g. ruler, protractor, bal-ance, calculator), they are encouraged to study the devel-opment of measuring systems and instruments or com-

COMPETENCY 2 • TO REASON USING MATHEMATICAL CONCEPTS AND PROCESSES


putational processes. They are asked to make a list ofmathematical processes and tools used in everyday lifeand in other school subjects so that they can betterunderstand them and appreciate their usefulness.

As with everything they study in elementary school, stu-dents will find it that much easier and more rewarding tolearn about mathematical reasoning and to becomefamiliar with the required mathematical concepts andprocesses if learning situations are made concrete oraccessible.


During Cycle One, the students work at putting togethera network of mathematical concepts and processes. Theystudy a few numerical patterns. They make connectionsbetween numbers and between operations and numbers.They identify easily observable geometric patterns anddevelop measurement sense so that they can describeand visualize their environment and move about in it.They engage in simple activities involving chance andinterpret and construct graphs representing variousaspects of their everyday life. They recognize situations intheir immediate environment that illustrate applicationsof mathematics and the usefulness of technology. Theyalso make connections between certain aspects of thehistory of mathematics and some of the concepts learned

in class. Discussions with classmates as well as the use oftechnology help the students explore and develop math-ematical concepts and processes.

During Cycle Two, the students develop their understand-ing of the number system. They describe and classify geo-metric objects according to their attributes. They con-struct more complex geometric relations and work withunconventional instruments and units of measure instudying surface areas and volumes. They continueexploring statistics and probability. By studying thehistory of mathematics, they make connections betweenthe needs of societies and the development of mathe-matics or technology. They become more familiar withmathematical terminology, symbolism, concepts andprocesses.

During Cycle Three, the students develop a greater under-standing of the meaning of numbers and operations.Theycontinue studying the attributes of geometric objects,constructing geometric relationships, exploring activitiesinvolving chance and interpreting statistical data. Thestudents identify situations in which mathematics canhelp them exercise critical judgment. They determinewhether it is appropriate to use technology in a givenactivity. They continue studying the links between thevarious needs of modern societies and certain mathe-matical discoveries. They consolidate their understandingof mathematical concepts and processes.



Evaluation Criteria– Appropriate analysis of a situation

involving applications ➊ ➋ ➌

– Choice of mathematical conceptsand processes appropriate to the givensituation involving applications ➊ ➋ ➌

– Appropriate application of thechosen processes ➊ ➋ ➌

– Correct justification of actions orstatements by referring to mathematicalconcepts and processes ➊ ➋ ➌


TO REASON

USING MATHEMATICAL

CONCEPTS AND PROCESSES

To mobilize mathematical concepts andprocesses appropriate to the given situa-tion

To apply mathematicalprocesses appropriate tothe given situation

To justify actions or statements byreferring to mathematical concepts andprocesses

To define the elements ofthe mathematical situation


CYCLE ONE

By the end of this cycle, the students devise and applytheir own processes to do mental and written computa-tions that involve adding and subtracting natural num-bers. They construct plane figures and solids and measurelengths and time using appropriate instruments and tech-nology.

CYCLE TWO

By the end of this cycle, the students continue developingand applying their own computational processes, but thistime they use the four operations. They become familiarwith conventional processes for written computationsthat involve adding and subtracting natural numbers anddecimals. They can describe plane figures and solids. Theybegin to estimate, measure or calculate lengths, surfaceareas and time. They can produce frieze patterns and tes-sellations by means of reflections. They can do simula-tions related to activities involving chance and interpretand draw broken-line graphs.Without really being able toexplain why, they can recognize situations in which it isappropriate to use technology.

CYCLE THREE

By the end of this cycle, the students mobilize their ownprocesses as well as conventional processes to do mentaland written computations involving the four operationswith natural numbers and decimals. Using objects anddiagrams, they start to add and subtract fractions and tomultiply fractions by natural numbers. They can describeand classify plane figures, recognize the nets for convexpolyhedrons, estimate, measure or calculate lengths, sur-face areas, volumes, angles, capacities, masses, time andtemperature. They can produce frieze patterns and tessel-lations by means of reflections and translations, comparethe possible outcomes of a random experiment with theknown theoretical probabilities, calculate the arithmeticmean and interpret circle graphs. They know how to jus-tify their use of technology.




The communication process benefits all those whoengage in any exchange of ideas, if only because the cir-culation of information is mutually rewarding. It is espe-cially useful to the person conveying a message becauseexplaining our understanding of a situation or a conceptoften helps us improve or deepen that understanding.

In the specific case of mathematics, we not only derivethe usual benefits associated with the communicationprocess, but also become familiar with mathematical lan-guage. The students interpret or produce a message (oralor written or in the form of a drawing) involving a line ofquestioning, an explanation or a statement related tomathematical activities dealing with arithmetic, geome-try, measurement, statistics and probability.

When communicating by using mathematical language,the students learn to name the processes and conceptsthey have studied in various activities, and this enablesthem to reinforce their understanding of these conceptsand processes. They observe how this language can beused to understand other subjects and everyday activi-ties. They sometimes study the development of this lan-guage throughout history.


When students pay attention to the accuracy and clarityof their mathematical message, to the medium used topresent their message and to the people to whom thismessage is addressed, they develop certain cross-curricu-lar competencies, in particular the ones relating to theability to communicate and to use information.


Students must communicate at different stages in thelearning process: when becoming familiar with a situa-tional problem they are asked to solve and when pre-senting possible solutions, comparing their points of viewor explaining their results. There are many examples ofthis type of communication: in arithmetic, for example,the students may be asked to formulate a situationalproblem that their classmates will have to solve. In geom-etry, they will make a scale drawing of a house thatsomeone else will have to build.


During Cycle One, the students become familiar with themeaning of certain mathematical terms and symbols andlearn how to use them to express their ideas and com-ment on those of others. During Cycle Two, they con-tinue learning about mathematical language by making agreater effort to distinguish between the meanings ofterms and symbols and by referring to different informa-tion sources. They take part in discussions with theirclassmates and compose simple messages. During CycleThree, they refine their choice of the mathematical termsand symbols they use to communicate information andcan give more precise explanations of their differentmeanings. They compare information from varioussources. In discussions with classmates, they make con-nections between other points of view and their ownopinions and adjust their message, if necessary.

COMPETENCY 3 • TO COMMUNICATE BY USING MATHEMATICAL LANGUAGE



CYCLE ONE

By the end of this cycle, the students interpret or producea message (oral or written) such as a statement, process,or solution by using simple mathematical language andat least one of the following types of representations:objects, drawings, tables, graphs, symbols or words.

CYCLE TWO

By the end of this cycle, the students interpret or producea message (oral or written) by using elaborate mathe-matical language and more than one type of representa-tion, including diagrams.

CYCLE THREE

By the end of this cycle, the students interpret or producea message (oral or written) by using exact mathematicallanguage and several types of representations.


Evaluation Criteria– Correct interpretation of a

message (oral or written) usingmathematical language ➊ ➋ ➌

– Correct production of a message(oral or written) usingmathematical language ➊ ➋ ➌


TO COMMUNICATE BY

USING MATHEMATICAL

LANGUAGE

To make connectionsbetween mathematicallanguage and everydaylanguage

To interpret or produce mathematicalmessages

To become familiar with math-ematical vocabulary


Essential KnowledgesAlthough science and technology are not part of the Cycle One program, many basiclearnings must be covered during this cycle in the other subjects. The study of mathe-matics provides an ideal opportunity to acquire these learnings because of its connec-tions with science and technology.

As with the rest of the essential knowledges, the study of measurement helps the stu-dents acquire the competency developed in Cycle One science and technology. As partof an introduction to the international system of measurement, measurements can, forexample, be used in collecting information during scientific-style experiments, in makinga simple technological object such as scale drawing of the classroom or a lever, or inmaking cutouts.

Because the mathematical concepts covered at the elementary level are connected toreal-world situations, there are many opportunities to examine the mathematical, scien-tific and technological aspects of a learning situation simultaneously.

LEARNING AND STRATEGY

ARITHMETIC: UNDERSTANDING AND WRITING NUMBERS

• Natural numbers

– natural numbers less than 1000 (units, tens, hundreds): reading,writing, digit, number, counting, one-to-one correspondence,representation, comparison, classification, order, equivalentexpressions, writing numbers in expanded form, patterns,properties (even numbers, odd numbers), number line ➊

– natural numbers less than 100 000 (thousands, ten thousands):reading, writing, representation, comparison, classification, order,equivalent expressions, writing numbers in expanded form,patterns, properties (squares, prime and compound numbers),number line ➋

– natural numbers less than 1 000 000 (hundred thousands): reading,writing, representation, comparison, classification, order, equivalentexpressions, writing numbers in expanded form, patterns, numberline ➌

– power, exponent ➌

– approximation ➊ ➋ ➌

• Fractions

– fractions related to the student’s everyday life ➊

– fractions based on a whole or a collection of objects: reading,writing, numerator, denominator, various representations(using objects or pictures), equivalent parts, comparison with 0,12 and 1 ➋

– fractions: reading, writing, numerator, denominator, variousrepresentations, order, comparison, equivalent expressions,equivalent fractions ➌

– percentages ➌

• Decimals

– up to two decimal places (tenths, hundredths): reading, writing,various representations, order, equivalent expressions, writingnumbers in expanded form ➋

– up to three decimal places (tenths, hundredths, thousandths):reading, writing, various representations, order, equivalentexpressions, writing numbers in expanded form ➌

– approximation ➋ ➌

• Using numbers

– converting from one type of notation to another: writing fractions,decimal numbers or percentages ➌

– choosing the most suitable notation for a given context ➌


ARITHMETIC: UNDERSTANDING AND WRITING NUMBERS (cont.)

• Integers

– reading, writing, comparison, order, representation ➌

ARITHMETIC: MEANING OF OPERATIONS INVOLVING NUMBERS

• Natural numbers

– operation, operation sense: addition (adding, uniting, comparing),sum, subtraction (taking away, complement, comparing), difference,term, missing term, number line, multiplication (repeated addition,Cartesian product) and division (repeated subtraction, sharing,number of times x goes into y) ➊

– choice of operation: addition, subtraction ➊

– operation sense: multiplication (e.g. repeated addition, Cartesianproduct), product, factor, multiples of a natural number, division(repeated subtraction, sharing, number of times x goes into y),quotient, remainder, dividend, divisor, set of divisors of a naturalnumber, properties of divisibility ➋ ➌

– choice of operation: multiplication, division ➋ ➌

– meaning of an equality relation (equation), meaning of anequivalence relation ➊ ➋ ➌

– relationships between the operations ➊ ➋ ➌

– property of operations: commutative law ➊

– property of operations: associative law ➋

– property of operations: distributive law ➌

– order of operations (series of operations involving natural numbers) ➌

• Decimals

– operation sense: addition and subtraction ➋

– operation sense: multiplication and division ➌

• Fractions

– operation sense (using objects and diagrams): addition, subtractionand multiplication by a natural number ➌

ARITHMETIC: OPERATIONS INVOLVING NUMBERS

• Natural numbers

– approximating the result of an operation: addition, subtraction ➊

– approximating the result of an operation: addition, subtraction,multiplication, division ➋ ➌

– own processes for mental computation: addition, subtraction ➊

– own processes for mental computation: addition, subtraction,multiplication, division ➋ ➌

– operations to be memorized:

- additions (0 + 0 to 10 + 10) related to the corresponding

subtractions ➊

- multiplications (0 × 0 to 10 × 10) related to the correspondingdivisions ➋

– own processes for written computation: addition, subtraction ➊

– own processes for written computation: multiplying a three-digitnumber by a one-digit number ➋

– own processes for written computation: dividing a three-digitnumber by a one-digit number ➋

– conventional processes for written computation: adding twofour-digit numbers ➋

– conventional processes for written computation: subtractinga four-digit number from a four-digit number such that thedifference is greater than 0 ➋

– conventional processes for written computation: multiplying athree-digit number by a two-digit number ➌


ARITHMETIC: OPERATIONS INVOLVING NUMBERS (cont.)

– conventional processes for written computation: dividing afour-digit number by a two-digit number, expressing the remainderas a decimal that does not go beyond the second decimal place ➌

– series of operations in accordance with the order of operations ➌

– patterns: series of numbers, family of operations ➊ ➋ ➌

– finding prime factors ➋ ➌

– divisibility by 2, 3, 4, 5, 6, 8, 9, 10 ➌

• Decimals

– approximating the result of an operation ➋ ➌

– mental computation: addition, subtraction ➋

– mental computation: addition, subtraction, multiplication, division ➌

– written computation: addition, subtraction; the result must notgo beyond the second decimal place ➋

– written computation: multiplication whose product does not gobeyond the second decimal place, division by a natural numberless than 11 ➌

– mental computation: multiplication and division of decimalsby 10, 100, 1000 ➌

• Fractions

– establishing equivalent fractions ➌

– reducing fractions, irreducible fractions ➌

– adding fractions using objects and diagrams, when thedenominator of one fraction is a multiple of the denominatorof the other fraction ➌

– subtracting fractions using objects and diagrams, when thedenominator of one fraction is a multiple of the denominatorof the other fraction ➌

– multiplying a natural number by a fraction, using objectsand diagrams ➌

GEOMETRY: GEOMETRIC FIGURES AND SPATIAL SENSE

• Space

– locating objects and getting one’s bearings in space, spatialrelationships (e.g. in front, on, to the left) ➊

– locating objects on an axis ➊ ➋ ➌

– locating objects in a plane ➊ ➋

– locating objects in a Cartesian plane ➋ ➌

• Solids

– comparing and constructing prisms, pyramids, spheres,cylinders, cones ➊

– comparing objects in the environment with solids ➊

– attributes (number of faces, base): prisms, pyramids ➊

– describing prisms and pyramids in terms of faces, vertices andedges ➋

– nets for prisms and pyramids ➋

– classification of prisms and pyramids ➋

– recognizing nets for convex polyhedrons ➌

– testing Euler’s theorem (relationship between faces, vertices andedges of a convex polyhedron) ➌

• Plane figures

– comparing and constructing figures made with closed curved linesor closed straight lines ➊

– identifying a square, rectangle, triangle, circle and rhombus ➊

– describing a square, rectangle, triangle and rhombus ➊

– describing convex and nonconvex polygons ➋

– describing quadrilaterals, including trapezoids and parallelograms:parallel segments, perpendicular segments, right angles,acute angles, obtuse angles ➋

GEOMETRY: GEOMETRIC FIGURES AND SPATIAL SENSE (cont.)– classifying quadrilaterals ➋

– constructing parallel lines and perpendicular lines ➋

– describing triangles: right triangles, isosceles triangles,scalene triangles, equilateral triangles ➌

– classifying triangles ➌

– measuring angles in degrees using a protractor ➌

– studying the features of a circle: radius, diameter, circumference,central angle ➌

• Frieze patterns and tessellations

– observing and producing patterns using geometric figures ➋

– congruent figures ➊

– observing and producing (grids, tracing paper) frieze patternsby means of reflections: reflection, line of reflection ➋

– observing and producing tessellations by means of reflections ➋

– observing and producing (grids, tracing paper) frieze patternsby means of translations: translation, translation arrow(length, direction, sense) ➌

– observing and producing tessellations by means of translations ➌

MEASUREMENT

• Lengths: estimating and measuring

– dimensions of an object ➊

– unconventional units: comparison, construction of rulers ➊

– conventional units (m, dm, cm) ➊

– conventional units (m, dm, cm, mm) ➋

– conventional units (km, m, dm, cm, mm) ➌

– relationships between units of measure ➋ ➌

– perimeter, calculating the perimeter ➋

• Angles: estimating and measuring

– comparing angles (right, acute, obtuse) ➋

– degree ➌

• Surface areas: estimating and measuring

– unconventional units ➋

– conventional units (m2, dm2, cm2), relationships between the unitsof measure ➌

• Volumes: estimating and measuring

– unconventional units ➋

– conventional units (m3, dm3, cm3), relationships between the unitsof measure ➌

• Capacities: estimating and measuring

– unconventional units ➌

– conventional units (L, mL), relationships between the unitsof measure ➌

• Masses: estimating and measuring

– unconventional units ➌

– conventional units (kg, g), relationships between the unitsof measure ➌

• Time: estimating and measuring

– conventional units, duration (day, hour, minute, second, daily cycle,weekly cycle, yearly cycle) ➊ ➋

– relationships between the units of measure ➌

• Temperatures: estimating and measuring

– conventional units (°C) ➌



STATISTICS

• Formulating questions for a survey ➊ ➋ ➌

• Collecting, describing and organizing data using tables ➊ ➋ ➌

• Interpreting data using a bar graph, a pictograph and a data table ➊

• Displaying data using a bar graph, a pictograph and a data table ➊

• Interpreting data using a broken-line graph ➋

• Displaying data using a broken-line graph ➋

• Interpreting data using a circle graph ➌

• Arithmetic mean (meaning, calculation) ➌

PROBABILITY

• Experimentation with activities involving chance ➊ ➋ ➌

• Predicting the likelihood of an event (certainty, possibilityor impossibility) ➊ ➋ ➌

• Enumerating the possible outcomes of a simple randomexperiment ➊

• Probability that a simple event will occur (more likely, just as likely,less likely) ➋ ➌

• Enumerating the possible outcomes of a random experimentusing a table, a tree diagram ➋ ➌

• Comparing the outcomes of a random experiment with knowntheoretical probabilities ➌

• Doing simulations with or without a computer ➋ ➌

Cultural References• Numbers

– origin and creation of numbers ➊

– development of systems for writing numbers ➊

– number systems (e.g. Arabic, Roman, Babylonian, Mayan):characteristics, advantages and disadvantages ➋ ➌

– social context (e.g. price, date, telephone, address, age, quantity:mass, size) ➊ ➋ ➌

• Operations

– own or conventional computation processes: development,limitations, advantages and disadvantages ➊ ➋ ➌

– technology: development (e.g. sticks, strokes, abacus, calculator,software), limitations, advantages and disadvantages ➊ ➋ ➌

– symbols (origin, development, need, mathematiciansinvolved): +, –, >, <, = ➊

– symbols (origin, development, need, mathematiciansinvolved): ∞, ÷, ≠ ➋

– interdisciplinary or social context (e.g. history, geography,science and technology) ➊ ➋ ➌

Geometric figures

– interdisciplinary or social context (e.g. architecture, maps, arts,decoration) ➊ ➋ ➌

– symbols (origin, development, need, mathematiciansinvolved): ∠, //, ⊥ ➋ ➌

• m, dm, cm, mm ➋

• km, m, dm, cm, mm ➌

• kg, g, L, mL ➌

• h, min, s (representation of time of day: 02:00, 2:00 a.m;representation of elapsed time: 2 h 10 min, 2:10) ➊ ➋ ➌

• °C ➌

• $, ¢ ➋ ➌

VOCABULARY

➊


• Measurement

– systems of measurement (historical aspect) ➊ ➋ ➌

– units of measure: development according to society’s needs(e.g. agrarian measurements, astronomy, standard measurement,precision), instruments (rudimentary approach formeasuring time, hourglass, clock) ➋ ➌

– symbols (origin, development, need): m, dm, cm ➊

– symbols (origin, development, need): m, dm, cm, mm ➋

– symbols (origin, development, need): km, m, dm, cm, mm ➌

– symbols (origin, development, need): kg, g, L, mL ➌

– symbols (origin, development, need): h, min, s ➊ ➋ ➌

– symbols (origin, development, need): °C ➌

– symbols (origin, development, need, mathematicians involved): ( ), % ➌

In each cycle, students in a given class carry out at least one individual or group projector activity related to cultural references.

SYMBOLS• 0 to 9, +, −, >, <, = ➊

• 0 to 9, +, −, ×, ÷, >, <, =, ≠ ➋

• 0 to 9, +, −, ×, ÷, >, <, =, ≠, ( ), % ➌

• Calculator keys [keys: 0 to 9, +, –, ×, ÷, =, ON, OFF, AC, C, CE(all clear, clear, clear last entry)] ➊ ➋ ➌

• Certain commonly used calculator functions [memories(M+, M–, MR, MC), change of sign (+/–)] ➌

• Numbers written using digits ➊ ➋ ➌

• Writing fractions ( ) ➊ ➋ ➌

• Writing decimals using a period as the decimal marker ➋ ➌

• Exponential notation �2, �3 ➌

• ∠, //, ⊥ ➋ ➌

• m, dm, cm ➊

ab

additionas many asas much as

bar graphbase of a solid

centimetrecertain eventchancecircleconecubecurved linecylinder

daydecimetredecreasing orderdepthdifferencedigit

even number

facefewerfraction

grouping

halfheighthourhundreds place

impossible eventincreasing order…is bigger than……is equal to……is smaller than…

lengthless

metreminusminutemore

natural numbernonenumbernumber line

odd numberone thirdone-to-one correspondence

pictographplane figurepluspossible eventprismprobable outcomepyramid

quarter

rectanglerhombus


VOCABULARY

➊

secondseriessidesolidspheresquarestraight linesubtractionsumsurvey

tabletens placetriangle

unitunit of measure

width

➋

acute angleangleareaat leastat most

base tenbroken-line graph

Cartesian planechance (statistics)compound numberconvex polygoncurved bodycurved surface

daily cycledecimaldenominatordividenddividing updivisiondivisor

edgeequalityequationequivalent partevent

factorflat surfacefrieze pattern

gram

hundredth

inequalityinverse operationparallelogram…is greater than……is less than……is not equal to……is parallel to……is perpendicular to…

just as likely

kilogram

less likelyline of reflection

measuring instrumentmillimetremissing termmore likelymultiplemultiplication

net of a solidnonconvex polygonnumber squarednumerator

obtuse angleordered pair

perimeterplace valueplanepolygonprime numberproduct

quadrilateralquotient

reference systemreflectionremainderright angle

segmentsurfacesymmetric figure

ten thousands placetenthtermtessellationthousandthousands placetrapezoidtree diagram

vertexvolume

weekly cyclewhole

yearly cycle

arithmetic mean

capacitycentral anglecircle graphcircumferenceconvex polyhedroncube of (the)

degree (angle)degree Celsiusdiameter

equilateral triangleequivalent fractionEuler’s theoremexponent

hundred thousands place

integerirreducible fractionisosceles triangle

kilometre

litre

massmillilitremillion

negative number

parenthesis

percentagepolyhedronpositive numberpowerprotractor

radiusright triangle

scalene trianglesquare of (the)

thousandthtranslationtranslation arrow

➌


Suggestions for Using Informationand Communications Technologies*• Becoming familiar with the basic operations of a calculator

[keys: 0 to 9, +, −, ×, ÷, =, ON, OFF, AC, C, CE (all clear, clear, clearlast entry), functions: recursive with the = key] ➊ ➋ ➌

• Becoming familiar with certain commonly used calculatorfunctions [memories (M+, M–, MR, MC), change of sign (+/–)] ➌

• Using technology for operations involving numbers that go beyondthe scope of the material covered in these cycles ➊ ➋ ➌

• Using technology to present proofs related to operations ➊ ➋ ➌

• Using a calculator in applying different problem-solving strategies ➊ ➋ ➌

• Using a calculator and a computer to explore natural numbersand operations ➊ ➋ ➌

• Using a calculator and a computer to explore decimals, fractionsand operations ➋ ➌

• Using a calculator and a computer to explore integers ➌

• Using a computer (graphics and spreadsheet software as wellas simulations) in applying different problem-solving strategies ➊ ➋ ➌

• Using a computer (word-processing, graphics and spreadsheetsoftware) to present information related to the solution ➊ ➋ ➌

• Producing a drawing (solids, plane figures, frieze patterns andtessellations) using graphics software ➊ ➋ ➌

• Using a computer to look for information ➋ ➌

• Learning to collect data using spreadsheet software ➋ ➌

• Learning to produce a graphic representation of data usingspreadsheet software ➋ ➌

• Learning to do a computer simulation of a random experiment ➋ ➌

• Using the Internet to find historical accounts related toconcepts studied in class ➋ ➌

• Consulting Internet Web sites on mathematics as well as glossariesand databases ➋ ➌

• Using interactive mathematics sites ➌

* The use of information and communications technologies is compulsory, but it is up to theteacher to choose the activities involving ICT.

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